What Is The Value Of Sec 3pi/4?
The value of sec 3pi/4 is $-\sqrt{2}$, or about $-1.4142$ as a decimal. In symbols, $\sec\frac{3\pi}{4} = -\sqrt{2}$.
The angle is written in radians as $\frac{3\pi}{4}$ and in degrees as $135^\circ$. Both name the same angle. Since one full turn is $2\pi$ radians or $360^\circ$, three-quarters of a half-turn lands at $135^\circ$, three-eighths of the way around the circle.
Secant is the reciprocal of cosine. That single fact drives the whole answer:
$$\sec\theta = \frac{1}{\cos\theta} \qquad \Rightarrow \qquad \sec\frac{3\pi}{4} = \frac{1}{\cos\frac{3\pi}{4}}$$
Because $\cos\frac{3\pi}{4} = -\frac{\sqrt{2}}{2}$, taking the reciprocal gives $-\frac{2}{\sqrt{2}} = -\sqrt{2}$. The rest of this article shows where that cosine comes from, why the sign is negative, and how to reach the value without memorising it.
How Do You Find Sec 3pi/4?
You find sec 3pi/4 in three short moves: fix the reference angle, fix the sign from the quadrant, then take the reciprocal of cosine.
Find the reference angle. The reference angle is the acute angle between the terminal side and the x-axis. For $\frac{3\pi}{4}$ in Quadrant II, it is $\pi - \frac{3\pi}{4} = \frac{\pi}{4}$, or $45^\circ$.
Fix the sign from the quadrant. Use ASTC ("All Students Take Calculus"). In Quadrant II only sine and its reciprocal are positive, so cosine and secant are negative here.
Take the reciprocal of cosine. At the reference angle, $\cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}$. Apply the Quadrant-II sign, then flip:
$$\cos\frac{3\pi}{4} = -\cos\frac{\pi}{4} = -\frac{\sqrt{2}}{2}, \qquad \sec\frac{3\pi}{4} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2}$$
The same value follows from the right triangle behind the reference angle. A $45^\circ$ right triangle has legs of equal length $1$ and hypotenuse $\sqrt{2}$, so $\cos 45^\circ = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}}$ and $\sec 45^\circ = \frac{\text{hypotenuse}}{\text{adjacent}} = \sqrt{2}$. Quadrant II turns that $\sqrt{2}$ into $-\sqrt{2}$.
Where Does 3pi/4 Sit On The Unit Circle?
On the unit circle, the angle $\frac{3\pi}{4}$ lands at the point $\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$, up and to the left of the centre. The x-coordinate is the cosine and the y-coordinate is the sine.
Secant reads straight off that x-coordinate, because $\sec\theta = \frac{1}{\cos\theta} = \frac{1}{x}$. Here $x = -\frac{\sqrt{2}}{2} \approx -0.7071$, and its reciprocal is $-\sqrt{2} \approx -1.4142$. The negative x-coordinate is exactly why the secant is negative.
How Do You Derive Sec 3pi/4 Exactly?
The exact form $-\sqrt{2}$ is not a rounded decimal, it comes out of an exact identity. Two clean derivations reach it.
Angle-difference on cosine. Write $\frac{3\pi}{4}$ as $\pi - \frac{\pi}{4}$ and use $\cos(\pi - \theta) = -\cos\theta$:
$$\cos\frac{3\pi}{4} = \cos\left(\pi - \frac{\pi}{4}\right) = -\cos\frac{\pi}{4} = -\frac{\sqrt{2}}{2}$$
$$\sec\frac{3\pi}{4} = \frac{1}{\cos\frac{3\pi}{4}} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}} = -\sqrt{2}$$
The last step rationalises $\frac{2}{\sqrt{2}}$: multiply top and bottom by $\sqrt{2}$ to get $\frac{2\sqrt{2}}{2} = \sqrt{2}$, then carry the negative sign.
Co-function check. Secant and cosecant are co-functions: $\sec\theta = \csc\left(\frac{\pi}{2} - \theta\right)$. Substituting $\theta = \frac{3\pi}{4}$:
$$\sec\frac{3\pi}{4} = \csc\left(\frac{\pi}{2} - \frac{3\pi}{4}\right) = \csc\left(-\frac{\pi}{4}\right) = -\csc\frac{\pi}{4} = -\sqrt{2}$$
Both routes agree on $-\sqrt{2}$, which is the sign that the value is exact and not an artefact of one method.
What Is The Secant Value Table Around 3pi/4?
Seeing the neighbours makes the sign flip and the undefined point obvious. Secant follows cosine: where cosine is positive, secant is positive; where cosine is zero, secant is undefined.
Table: Secant across the top of the unit circle, from $0$ to $\pi$.
Angle | Radians | $\cos$ | $\sec$ |
|---|---|---|---|
$0^\circ$ | $0$ | $1$ | $1$ |
$45^\circ$ | $\frac{\pi}{4}$ | $\frac{\sqrt{2}}{2}$ | $\sqrt{2} \approx 1.4142$ |
$90^\circ$ | $\frac{\pi}{2}$ | $0$ | undefined |
$135^\circ$ | $\frac{3\pi}{4}$ | $-\frac{\sqrt{2}}{2}$ | $-\sqrt{2} \approx -1.4142$ |
$180^\circ$ | $\pi$ | $-1$ | $-1$ |
Notice the symmetry. The secant at $\frac{\pi}{4}$ and at $\frac{3\pi}{4}$ have the same size, $\sqrt{2}$, and opposite signs. That mirror is the reference angle $\frac{\pi}{4}$ doing its work on both sides of the vertical. For the full grid of standard angles, see the trigonometric table.
Why Is Sec 3pi/4 Negative?
The negative sign is not a rule to memorise. It follows from where the angle points.
The angle is in Quadrant II. At $135^\circ$ the terminal side points up and to the left, so its horizontal reach is to the left of centre. On the unit circle that horizontal reach is the x-coordinate, and it is negative.
Cosine is that x-coordinate. A negative x-coordinate means $\cos\frac{3\pi}{4} < 0$.
Secant is the reciprocal of cosine. One divided by a negative number stays negative, so $\sec\frac{3\pi}{4} < 0$ too. The reciprocal changes the size, never the sign.
ASTC confirms it. In Quadrant II only sine and cosecant are positive. Cosine, secant, tangent, and cotangent are all negative there.
So the value has to be a negative version of the reference-angle secant $\sqrt{2}$, which is exactly $-\sqrt{2}$. For more on how the three reciprocal functions track their parents, see reciprocal identities.
Who Discovered The Secant Function?
Secant is younger than sine, and its story runs through three continents. The idea of relating an angle to a length in a circle is ancient, but "secant" as a named function is a much later, medieval-into-Renaissance invention.
Two more figures shaped the path to the modern function:
Aryabhata (476–550 CE, India) compiled an influential table of half-chords, the jya, which became our sine after a long chain of translation. His work made the ratio-based view of these functions possible.
Thomas Fincke (1561–1656, Denmark) is credited with popularising the terms "secant" and "tangent" in his 1583 Geometria rotundi, fixing the vocabulary the world still uses.
Where Is Sec 3pi/4 Used In The Real World?
Secant appears wherever a horizontal distance stretches as an angle tilts, and the second-quadrant version shows up whenever that tilt passes the vertical.
Optics and light: in the physics of refraction and beam spread, secant terms describe how far a beam reaches across a surface as its angle steepens, which is the searchlight idea from the top of this article.
Architecture and construction: the length of a rafter or a ramp compared with its horizontal run is a secant relationship, so roof pitch and slope calculations lean on it directly.
Surveying and navigation: sighting a distant point above or across a baseline uses secant to convert a measured angle into a real distance.
Computer graphics and cameras: a camera's field of view and the projection of a scene onto a flat screen involve secant-like scaling as the viewing angle widens.
Across every one of these, the sign tells direction. A negative secant, the kind $\frac{3\pi}{4}$ produces, is the mathematics quietly recording that something has tipped past its upright reference.
What Are The Most Common Mistakes With Sec 3pi/4?
These four errors account for most lost marks on secant values, verified against reference-angle guides on Mathwords and Purplemath and the standard ASTC sign convention.
Dropping the negative sign.
Where it slips in:
A student finds the reference-angle secant $\sqrt{2}$ and writes that as the answer, forgetting that $\frac{3\pi}{4}$ sits in Quadrant II.
Don't do this:
Do not report $\sqrt{2}$. The reference angle gives only the size, never the sign.
The correct way:
Apply the quadrant sign before finishing. In Quadrant II secant is negative, so $\sec\frac{3\pi}{4} = -\sqrt{2}$.
Measuring the reference angle to the wrong axis.
Where it slips in:
A student measures the reference angle up to the y-axis and gets $\frac{\pi}{4}$ from the top instead of from the horizontal, sometimes landing on the wrong acute angle entirely.
Don't do this:
Do not measure to the y-axis. The reference angle is always taken to the nearest part of the x-axis.
The correct way:
From the x-axis, the reference angle is $\pi - \frac{3\pi}{4} = \frac{\pi}{4}$. Measuring to the horizontal keeps the reference angle consistent across all four quadrants.
Leaving the calculator in the wrong mode.
Where it slips in:
A student types $\sec(3\pi/4)$ with the calculator set to degrees, so the machine reads the input as $135^\circ$ worth of degrees times $\pi$ and returns nonsense.
Don't do this:
Do not evaluate a radian expression in DEGREE mode, and remember most calculators have no direct "sec" button.
The correct way:
Switch to RADIAN mode, compute $\cos(3\pi/4)$, then take its reciprocal with the $1/x$ key. The screen should read about $-1.4142$.
Confusing secant with cosine instead of its reciprocal.
Where it slips in:
A student reports $\cos\frac{3\pi}{4} = -\frac{\sqrt{2}}{2}$ as the secant, stopping one step early.
Don't do this:
Do not hand in the cosine. Secant is one divided by cosine, not cosine itself.
The correct way:
Finish the reciprocal: $\sec\frac{3\pi}{4} = \dfrac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2}$.
Practice Problems On Sec 3pi/4
Work each one, then check against the answer.
State $\sec\frac{3\pi}{4}$ in exact form.
(Answer: $-\sqrt{2}$.)Give $\sec\frac{3\pi}{4}$ as a decimal to four places.
(Answer: $-1.4142$.)Find $\cos\frac{3\pi}{4}$ and then its reciprocal.
(Answer: $\cos\frac{3\pi}{4} = -\frac{\sqrt{2}}{2}$, reciprocal $-\sqrt{2}$.)What is the reference angle for $\frac{3\pi}{4}$, in radians and degrees?
(Answer: $\frac{\pi}{4}$, or $45^\circ$.)Compare $\sec\frac{\pi}{4}$ and $\sec\frac{3\pi}{4}$.
(Answer: $\sqrt{2}$ and $-\sqrt{2}$, same size, opposite sign.)Evaluate $\sec\frac{3\pi}{4} \times \cos\frac{3\pi}{4}$.
(Answer: $(-\sqrt{2}) \times \left(-\frac{\sqrt{2}}{2}\right) = \frac{2}{2} = 1$, since a function times its reciprocal is $1$.)
Where Should You Go Next After Sec 3pi/4?
One value opens onto the whole reciprocal family and the circle it lives on.
Cos 3pi/4. The cosine underneath this whole page, derived on its own so the reciprocal step is fully grounded.
Sec pi/4. The positive mirror value, useful for seeing exactly what the Quadrant-II sign changes and what it leaves alone.
Unit circle with tangent. The circle that makes every one of these values visible at once, with the coordinates that generate them.
If your child is building these foundations, a live Bhanzu trainer teaches secant values starting from the unit circle and the "why" behind the sign, not from a table to memorise, in the Bhanzu trigonometry program.
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